Implications of Design Assumptions on Capacity ...

Implications of Design Assumptions on Capacity Estimates and Demand Predictions of Multispan Curved Bridges Aman Mwafy1; Amr Elnashai, F.ASCE2; and W.-H. Yen3 Abstract: The paper presents a detailed seismic performance assessment of a complex bridge designed as a reference application of modern codes for the Federal Highway Administration. The assessment utilizes state-of-the-art assessment tools and response metrics. The impact of design assumptions on the capacity estimates and demand predictions of the multispan curved bridge is investigated. The level of attention to detail is significantly higher than can be achieved in a mass parametric study of a population of bridges. The objective of in-depth assessment is achieved through investigation of the bridge using two models. The first represents the bridge as designed 共including features assumed in the design process兲 while the second represents the bridge as built 共actual expected characteristics兲. Three-dimensional detailed dynamic response simulations of the investigated bridge, including soil-structure interaction, are undertaken. The behavior of the as-designed bridge is investigated using two different analytical platforms for elastic and inelastic analysis, for the purposes of verification. A third idealization is adopted to investigate the as-built bridge’s behavior by realistically modeling bridge bearings, structural gaps, and materials. A comprehensive list of local and global, action and deformation performance indicators, including bearing slippage and inter-segment collision, are selected to monitor the response to earthquake ground motion. The comparative study has indicated that the lateral capacity and dynamic characteristics of the as-designed bridge are significantly different from the as-built bridge’s behavior. The potential of pushover analysis in identifying structural deficiencies, estimating capacities, and providing insight into the pertinent limit state criteria is demonstrated. Comparison of seismic demand with available capacity shows that seemingly conservative design assumptions, such as ignoring friction at the bearings, may lead to an erroneous and potentially nonconservative response expectation. The recommendations assist be given to design engineers seeking to achieve realistic predictions of seismic behavior and thus contribute to uncertainty reduction in the ensuing design. DOI: 10.1061/共ASCE兲1084-0702共2007兲12:6共710兲 CE Database subject headings: Bridges, continuous; Bridges, spans; Curvature; Three-dimensional models; Nonlinear analysis; Bearings; Friction resistance; Earthquake loads.

Introduction Multispan highway bridges are strategic elements in modern transportation networks that require careful design to ensure their functionality during and after earthquakes. The reliable detailed assessment of such complex structures is thus vital for deciding whether their design procedures are adequate. Such reliable assessment will also enable improvement in seismic performance by allowing for timely intervention. There are two bounding approaches for studying bridge structures: 共1兲 extensive parametric studies that require many parametric variations 共such as type of 1 Senior Research Fellow, Univ. of Illinois at Urbana-Champaign, Urbana, IL 61801. Assistant Professor 共on leave兲, Univ. of Zagazig, Egypt 共corresponding author兲. E-mail: [email protected] 2 Bill and Elaine Hall Endowed Professor of Civil Engineering, Director of the Mid-America Earthquake Center, Univ. of Illinois at Urbana-Champaign, Urbana, IL 61801. E-mail: [email protected] 3 Program Manager of Seismic Hazard Mitigation, Office of Infrastructure, R&D Fedral Highway Administration, McLean, VA 22101. E-mail: [email protected] Note. Discussion open until April 1, 2008. Separate discussions must be submitted for individual papers. To extend the closing date by one month, a written request must be filed with the ASCE Managing Editor. The manuscript for this paper was submitted for review and possible publication on February 2, 2006; approved on October 16, 2006. This paper is part of the Journal of Bridge Engineering, Vol. 12, No. 6, November 1, 2007. ©ASCE, ISSN 1084-0702/2007/6-710–726/$25.00.

foundation, characteristics of piers, pier-to-deck connections, deck section properties, abutment type and deck-abutment connection兲 and that are thus not extremely detailed; and 共2兲 extremely detailed analysis of a sample complex bridge that highlights many important issues of bridge system response. Both approaches are important and complementary. The current study belongs to the second class. The proportion of horizontal load to vertical load generally increases in continuous, joint-free, redundant structures. Hence, an efficient design and energy dissipation is attained by artificially increasing the period of vibration and the energy dissipated in secondary elements, thus maintaining the integrity of primary structural members. Base-isolation bridges exploit seismic isolation bearings and dissipation/damping devices to achieve this design philosophy. Elastomeric bearings can act as isolation devices to dissipate energy via their inelastic response and friction. Because primary structural members receive the most attention and movement at abutments is restricted, the bearing frictional resistance may be neglected in the design. This assumption may be nonconservative because polytetrafluoroethylene 共PTFE兲 bearings, which may have low friction at low velocity rates, generate higher friction under high seismic deformation 共Constantinou et al. 1990; Priestley et al. 1996兲. It is therefore necessary to investigate the assumptions conventionally adopted in the design and compare the ensuing behavior with more realistic simulations to assess their effects on the seismic integrity of bridge structures. The two key elements of assessment procedures are capacity

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Fig. 1. FHWA nine-span steel-girder bridge

and demand. Capacity 共supply兲 is a measure of the capability of the system to resist seismic actions, while the demand is a measure of the requirements imposed by earthquake ground motions. Pushover analysis is a powerful tool for evaluating lateral forceresisting capacity and may be employed to predict the seismic demand by estimating the target displacement at the design earthquake. Several improvements have been suggested in recent years to advance pushover analysis 共Elnashai 2001; Antoniou and Pinho 2004兲. However, newly developed procedures still do not guarantee satisfactory results in the case of increasing input ground motion peculiarity and structural irregularity 共Elnashai 2002兲. The number of enhancements involved in new proposals may also impact simplicity, an important criterion for analysis procedures intended for the design office environment. Although the conventional pushover procedure is more applicable to structures vibrating mainly in their fundamental mode 共Mwafy and Elnashai 2001兲, or at best a single mode, it has proved valuable for capacity estimates of long period structures and highway bridges 共Mwafy et al. 2005; Zheng et al. 2003; Lu et al. 2004; Paraskeva et al. 2006兲. Regardless, more research is needed to investigate the applicability of pushover analysis for capacity and limit state predictions of multispan complex bridges, particularly those with curvature, sliders, and expansion joints. The objective of this study is to investigate the effects of the simplifying assumptions typically used in design on the dynamic characteristics and capacity-demand predictions of multispan complex bridges, using comparison of “as-designed” and “asbuilt” simulations. The study is conducted on a 1,488 ft. nine-span bridge carefully selected from the Federal Highway Administration 共FHWA兲 inventory to represent a typical U.S. design for a complex highway bridge 共FHWA 1996兲. Extensive inelastic pushover and response history analyses are performed using state-of-the-art analysis tools to verify the analytical models and to compare estimates of capacities and demands. The study focuses on the following subobjectives: 共1兲 to present the methodology adopted to assess the seismic response of complex bridges; 共2兲 to verify different modeling approaches and select relevant performance criteria; 共3兲 to investigate the applicability of pushover analysis for evaluation of capacities and controlling limit state criteria; and 共4兲 to compare the capacity with the demand of the design assumptions and the as-built configurations to

evaluate the impact of various modeling approaches on the seismic integrity of multispan bridges.

Structural Characteristics and Analytical Modeling The application case study investigated herein is a U.S.-designed multispan bridge constructed in a medium seismicity region with a peak ground acceleration 共PGA兲 of 0.15 g. A number of borings drilled along the bridge alignment indicated that the subsoil is comprised of coarse alluvial flood deposits 共very dense sand and gravel兲 for a depth of 15 m overlying volcaniclastic sediments. The plan and elevation of the 454-m nine-span bridge are shown in Fig. 1. The bridge consists of two units—a four-span straight unit and a five-span curved one—separated by an expansion joint. While the two units act independently in the longitudinal direction, they are linked in transverse deformation at the intermediate expansion joint and pivot at the abutments. The superstructure is composed of four steel girders with a composite cast-in-place concrete deck. Conventional steel-pinned bearings are used at Piers 1, 2, 3, 6, and 7 to resist the seismic forces in the longitudinal direction. The bearings at the abutments, the expansion joint, and Piers 5 and 8 are PTFE sliders, which provide restraint in the transverse direction only. Elastomeric bearings with stainless-steel sliding surfaces were employed for all movable bearings. The transverse resistance is provided via girder stops capable of transferring transverse forces. The seat-type abutments and the single-column intermediate piers are all cast-in-place and supported on steel H-piles. The bridge’s design conforms to the AASHTO Standard Specifications 共AASHTO 1995兲. Since the bridge crosses the main channel of a wide river, flow and ice loading dictated the dimensions of the piers. The case study thus exemplifies a bridge controlled by segment collision and deformation rather than by ductility and strength limit state criteria, as subsequently discussed. This emphasizes the significance of different performance indicators in seismic design and the assessment of bridges. Furthermore, in view of recent changes in the U.S. seismic hazard maps 共USGS 2002兲, this design example was selected to investigate the vulnerability of complex bridges in medium seismicity

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Fig. 2. Modeling approach of superstructure

regions under higher levels of ground motions than the design earthquake. Further structural detailing and geotechnical information are provided elsewhere 共FHWA 1996兲. Refined 3D models of the bridge, including foundations and soil effect, were assembled for elastic and inelastic analysis using SAP2000 共2003兲 and ZEUS-NL 共Elnashai et al. 2004兲, respectively. The former modeling approach was used to verify the ZEUS-NL fiber modeling with the design before executing the extensive inelastic analysis. ZEUS-NL was used mainly to estimate the capacities and demands from inelastic pushover and response history analysis. The latter finite element analysis platform was developed and thoroughly verified at Imperial College, U.K., and the University Illinois at Urbana-Champaign to deliver a state-of-the-art simulation platform for static and dynamic analysis of steel, concrete, and composite structures 共Jeong and Elnashai 2005兲. In the detailed ZEUS-NL modeling, each structural member is assembled using a number of cubic elastoplastic elements representing the spread of inelasticity within the member cross section and along the member length. Sections are discretized to steel and confined and unconfined concrete fibers. The stress-strain response at each fiber is monitored during the entire multistep analysis. Gravity loads and mass are distributed on the superstructure and along the height of the piers. The employed distributed mass elements use cubic shape function and account for both the translational and rotational inertia. Figs. 2 and 3 depict the ZEUS-NL modeling approach of the superstructure, piers, and foundation. The modeling approaches adapted to idealize the superstructure, foundation, and bearings are discussed below. Superstructure and Material Modeling Modern seismic design philosophy of bridges relies on the piers to dissipate energy rather than on the superstructure, which remains elastic under the design earthquake. Based on the conventional elastic theory, the superstructure is modeled using three different cross sections: two hollow steel box sections equivalent to the four steel girders and a RC rectangular cross section representing the deck. This conforms to the following two criteria: 共1兲 equal sectional areas and 共2兲 equivalent sectional moments of

inertia and torsional constant. It is important to note that the total torsional resistance of the employed closed steel cross sections is higher than that of the steel girders. However, the steel cross frames, employed to transfer the superstructure mass to the piers, increase the torsional resistance of the four steel girders. Thus, the adopted modeling approach realistically predicts the elastic behavior of the superstructure. The elements idealizing the superstructure are located at the centeroid of the cross sections and are connected using rigid arms, as shown in Fig. 2. Employing the characteristic material strength causes a reduction in stiffness and an elongation in period. Hence, the characteristics values are used only for modeling the design configuration, while the more realistic mean values are employed to assess the response of the as-built bridge. A normal distribution is adopted to estimate the concrete compressive strength and steel yield strength. An average coefficient of variation 共COV兲 of 12% for concrete, assuming average curing conditions and workmanship, and 6% for the steel, calculated from previous experiments 共Rossetto and Elnashai 2005兲, are adopted. Foundation Modeling Based on the boring profile at the construction site, a soil type I was used in the design 共AASHTO 1995兲. This corresponds to stable deposits of sand and gravel of less than 61 m overlying rock. The kinematic soil-structure interaction is accounted for by restraining the pile caps with grounded springs representing the stiffness of the bridge pile foundations, as depicted in Fig. 3共b兲 for an intermediate pier. The local coordinates of individual piers are used to idealize the foundation stiffnesses and are calculated as follows: • The axial and lateral stiffnesses of an individual pile were calculated and the pile group stiffnesses were obtained from combinations of individual piles. • As a result of the footing’s high rigidity, the pile cap’s contribution was neglected. This is justified by the large relative stiffness of the foundation compared with the pier. The resulting demand on the substructure did not change because this contribution was ignored. • Because the soil may not be in full contact with the pile caps,

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Fig. 3. ZEUS-NL model for nine-span bridge

and to account for any liquefaction potential, the soil contribution was conservatively disregarded in this modeling. The bridge’s location, in the flood plain of a large river, supports this scenario: contact with the soil was lost under the pile caps. In the pile-bent abutment type used in the design, wing walls did not contribute much to the transverse horizontal resistance. Therefore, the transverse horizontal stiffness of the abutment is estimated using the translational stiffness of the abutment pile group. In the longitudinal direction, the abutment embankment fill stiffness is estimated based on the findings of a large-scale abutment test 共Maroney 1995兲. The initial passive stiffness from this testing 共11.5 kN/ mm/ m兲 was adjusted relative to the back-wall height 共Caltrans 2004兲. A maximum passive pressure was recommended in the Caltrans study, allowing the back-wall to break off to prevent inelasticity in the foundation system. The stiffness in active action is assumed to be one-fifth of the passive stiffness 共Choi et al. 2004兲. A bilinear elastoplastic relationship is therefore adopted to model the longitudinal behavior of the abutment. The vertical translation and torsional rotation of the abutmentsuperstructure connection are fully restrained, while the rotation about the transverse and vertical axis is released. The springs and releases conform to the local coordinate system of the abutments. Bearings and Gaps Modeling A zero frictional resistance is initially assumed at all sliding bearings and movement restrictions are assumed at abutments. Although this scenario is rather unrealistic, it was employed to maintain consistency with the assumptions made in the design phase. A rational estimation of the bearing frictional resistance is adopted afterward, in another idealization, to assess the behavior

of the as-built structure. The PTFE-stainless-steel movable bearings used in the design have a small friction coefficient at the low velocity rates 共2–5%兲 but have higher friction under high seismic deformation. For non-lubricated bearings, this coefficient at high velocities ranges from 5 to 15% and even higher at the low temperature 共Constantinou et al. 1990; Priestley et al. 1996; Bondonet and Filiatrault 1997兲. Previous experimental studies have also concluded that the coefficient of friction slightly decreases again under high velocities caused by frictional heating. The bridge investigated in the current study was assessed under increasing levels of input ground motion. Accordingly, a 10% friction coefficient was used in the analytical model to assess the behavior of the bridge as built. Two modeling approaches were investigated to idealize the impact stiffness. In the first approach, the pounding is represented by a nonlinear spring 共Hertz model兲. The impact stiffness increases in this approach from 2.92E6 MPa to 4.38E6 MPa at a penetration of 1.27 mm and increases again at 2.54 mm to 8.76E6 MPa 共Choi et al. 2004兲. These values are controlled to ensure that the penetration of pounding is less than 2.54 mm 共0.1 in兲. Muthukumar and DesRoches 共2006兲 concluded that this modeling approach is sufficient for the impact simulation at a PGA of 0.1–0.3 g, which is the intensity range investigated in the present study. Results obtained from this idealization were compared with a more simplified linear spring model employing the maximum impact stiffness 共Ki兲 of 8.76E6 MPa. It was concluded that the former model has insignificant effect on the response at the design earthquake, while it has a marginal effect at twice the design earthquake, compared with the simplified linear model. Therefore, the simplified approach was used to model the impact

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Fig. 4. Force-displacement relationships of joint elements representing expansion joint and abutment gaps

stiffness. Fig. 4 shows the force versus relative displacement relationships of the joint elements representing the expansion joint, movable bearings, and structural gaps. In this modeling, a positive relative displacement corresponds to an opening of the joint gap and a negative displacement corresponds to a closing of the gap. When the gaps at the abutment and at the expansion joint undergo a relative movement in the negative direction 共joint close兲 exceeding the gap width, the joint element begins resisting further opening 共collisions兲. It is clear that slippage takes place in the as-built modeling when the applied force reaches the maximum friction developed on the contact plane of the bearing.

Limit States and Input Ground Motions The performance criteria for yield and failure are classified into two groups 共Mwafy and Elnashai 2001; Mwafy and Elnashai 2002兲: local 共member-level兲 and global 共structure-level兲. Member yield is considered when the strain in the main longitudinal tensile reinforcement exceeds the steel yield strain. The yield limit state on the structure level is estimated from an elastoplastic idealization of the lateral response. Exceeding the ultimate curvature is considered the local failure criterion. This is controlled by the confined concrete’s ultimate compression strain at the extreme fiber, which is estimated according to the level of confinement. The shear demand-supply response is considered satisfactory since the pier shear strength is significantly higher than the maximum demand at twice the design earthquake. By considering overall structural characteristics, the following criteria are utilized to define global failure: 共1兲 drift of 3.0%; 共2兲 degradation of lateral strength of more than 10%; 共3兲 slippage 共unseating兲 or collision failure at the abutments or the expansion joint; and 共4兲 formation of a hinging mechanism. The gap elements employed at the expansion joint and at the abutments allow for predicting possible unseating or pounding between different structural segments, as indicated in Fig. 4. To obtain complete capacity enve-

lopes of the structure and its piers, the gap elements are not included in pushover analysis. A set of artificially generated input ground motions was carefully selected to represent three earthquake scenarios in MidAmerica for this assessment study 共Rix et al. 2004兲. The synthetic records were generated for two different soil profiles: lowlands and uplands. An 84% amplification factor was used for generating ten time histories for each of the three earthquake scenarios. The similarity between these records allowed a single accelerogram to be selected from each soil-earthquake scenario group. Hence, six records were used in response history analysis. These are Lowland1, Lowland2, Lowland3, Upland1, Upland2, and Upland3. The earthquake scenarios are as follows: • Scenario 1: M⫽7.5 at Blytheville, Arkansas, with a focal depth of 10 km. • Scenario 2: M⫽6.5 at Marked Tree, Arkansas, with a focal depth of 10 km. • Scenario 3: M⫽5.5 at Memphis, Tennessee, with a focal depth of 20 km. Fig. 5 compares the elastic spectra of the selected accelerograms to the design spectrum 共AASHTO 1995兲. The spectra of the records generated for Scenario 1 reasonably match the design spectrum. They, particularly Lowland1, are slightly higher in the period range below 1.0 s and lower than the design spectrum in the long period range. The latter record is likely to produce the highest demands because its amplification is high at 1.7 s, and may coincide with the fundamental cracked period of the design configuration in the longitudinal direction 共T1 uncracked = 1.42兲. The PGAs of Scenario 1 records are also akin to the design 共0.15 g兲. Therefore, Scenario 1 records are used to assess the design’s acceptability, whereas records generated for Scenarios 2 and 3 are used to check the serviceability limit states. Furthermore, in view of recent changes in the seismic hazard levels of several medium seismicity regions in the U.S. 共USGS 2002兲, the case study bridge’s vulnerability to ground motions greater than the design earthquake is investigated by scaling the records generated for

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Fig. 5. Comparisons between selected earthquake scenarios and design spectrum

periods obtained from ZEUS-NL are slightly smaller than those from SAP2000 because the superstructure modeling approach adopted in the two programs is different and the reinforcement rebars, which are effectively idealized in the former model, caused an increase in stiffness. Following the design assumptions, the SAP2000 model employs an equivalent RC cross section, in which the cross-sectional area of the steel girders is transformed to a concrete section using the modular ratio. Also, the concrete deck was considered to estimate only the torsional rigidity. Since these assumptions underestimate the superstructure’s geometrical properties, the ZEUS-NL modeling approach employs a more rational idealization, as discussed earlier 共refer to Fig. 2兲. Therefore, the ZEUS-NL results are in principle more realistic compared with those obtained from SAP2000. For this modeling approach, which disregards friction, it is confirmed that the two units of the structure independently vibrate in the longitudinal direction. The stiffness of Unit 1 is higher than that of Unit 2 in this direction as a result of the participation of three piers in resisting the inertia forces transmitted from the superstructure. Comparison of the dynamic characteristics of the design assumptions and the as-built bridge models is shown in Fig. 6. It is clear that the bridge’s behavior is significantly and fundamentally affected by friction. The fundamental period decreases by about 50% and the modes of vibration are altered when a 5% frictional resistance is considered. Unlike the case without friction, the first mode is both longitudinal and transverse, whereas the second and subsequent modes are in the transverse direction only. Table 1 compares the periods obtained considering different bearing fric-

Scenario 1 to twice the design earthquake. This conservative assumption implies that the seismicity of the site has increased by 100%. Several approaches for scaling input ground motions have been suggested in the literature. A procedure based on the velocity spectrum intensity 共Housner 1952兲 and using the structure’s inelastic period was adopted by Elnashai and Mwafy 共2002兲 and Mwafy and Elnashai 共2001兲. In this approach the natural records are scaled to possess equal velocity spectrum intensity for the period range of the structure, while the design spectrum is taken as a reference. This approach is significant when employing natural ground motions in the analysis. The present study adopts a more simplified approach—scaling the accelerograms using PGAs—because the synthetic records of Scenario 1 have spectrum intensities comparable to the design spectrum’s. The adopted set of accelerograms and PGAs thus ensures that the investigated structure is analyzed under input ground motions representing a wide range of possible seismic events at the site.

Dynamic Characteristics and Model Verification Comparisons of the periods of the design configuration obtained from SAP2000 and ZEUS-NL are shown in Table 1. The periods and mode shapes of both analytical platforms are comparable. The first and second mode shapes are in the longitudinal direction for Units 2 and 1, respectively. The predominant mode in the transverse direction is the third mode 关refer to Fig. 6共a兲兴. The

Table 1. Comparison between Periods of Vibration of Different Analytical Modeling ZEUS-NL as-built configurationb ZEUS-NL design configurationa

Period

SAP2000 design configurationa 共sec.兲

Period 共sec.兲

Difference 共%兲

5% friction 共stiffness= K兲 Period 共sec.兲

Difference 共%兲

T1 1.518 1.422 6.3 0.796 47.6 T2 1.207 1.130 6.4 0.763 36.8 T3 0.802 0.786 2.0 0.709 11.6 T4 0.748 0.732 2.1 0.651 13.0 T5 0.748 0.699 6.6 0.609 18.6 Note: K: the elastic stiffness of the elastomeric pad. a Design configuration: Friction is neglected and characteristics values of material strength are used. b As-built configuration: Bearing friction and mean values of material strength are considered.

30% friction 共stiffness= K兲

30% friction 共stiffness= 6K兲

Period 共sec.兲

Difference 共%兲

Period 共sec.兲

Difference 共%兲

0.796 0.763 0.709 0.651 0.609

47.6 36.8 11.6 13.0 18.6

0.764 0.709 0.659 0.608 0.594

49.7 41.3 17.8 18.7 20.6

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Fig. 6. ZEUS-NL dynamic characteristics for 共a兲 bridge featuring assumptions considered in design; 共b兲 as-built configuration with bearing friction

tional resistance and elastic stiffness 共K兲. The elastic stiffness is calculated from the shear modulus 共G兲, plan area 共A兲, and total elastomer thickness 共t兲. Thus, K = G ⫻ A / t. Clearly, the dynamic characteristics are not influenced by an increase in the frictional resistance, but they are marginally affected by the elastic stiffness of the pad. The significant changes in dynamic characteristics reflect a need to investigate the effect of bearing friction on a bridge’s inelastic response. Response spectrum analysis is performed using the design response spectrum 共AASHTO 1995兲. Thirty-six modes of vibration are employed to reach a 90% mass participation in the two principle directions. The complete quadratic combination 共CQC兲 method, which accounts for the coupling between modes, is used to combine the modal forces and displacement. The results are similar to those used in the design, which lend extra weight to the analytical models developed in the current study and allow for comparisons between the design parameters obtained from SAP2000 elastic analysis and ZEUS-NL inelastic pushover and response history procedures. Shortcomings of the design modeling assumptions are exemplified from this simple analysis, particularly in the longitudinal direction. High base shear demands are attracted to Piers 1 and 7 共3,185 and 4,057 kN, respectively兲. Since these short piers are provided with pinned bearings, they attract higher seismic demands because of their higher stiffness. A nonuniform distribution of displacement demands between Units 1 and 2 is also observed since both units are uncoupled and vibrate independently.

Capacity Estimates and Prioritizing Limit States Modern seismic design codes and guidelines 共EC8 2004; FEMA 2003; NCHRP 12-49 2001兲 adopt static pushover analysis as a design and assessment tool. The sequence of yielding and failure, as well as the progress of the overall capacity curve of the structure, are traced in the current study for both the design and the as-built configurations. This enables the identification of potential structural deficiencies and the estimation of the lateral capacity and provides insight into the limit state criteria required for response history analysis. To represent the distribution of inertia forces imposed on the investigated bridge, lateral force profiles are calculated as a combination of load distributions obtained from eigenvalue analysis. A number of modes of vibration in the longitudinal and transverse directions are selected based on their

mass participation to calculate the load patterns in the two principal directions. The gaps at the abutments and the expansion joints 共refer to Fig. 4兲 are not modeled in this incremental analysis to allow reaching the ultimate capacity and to obtain complete capacity envelopes of the structure and its piers. Accordingly, the adopted target displacement corresponds to the drift collapse limit state. Capacity of Individual Piers The height of Piers 1, 7, and 8 is 15 m, but it is 21 m for other intermediate piers, as shown from Fig. 3. Pushover analysis is thus carried out on each of the two pier configurations in the longitudinal and transverse directions to estimate their lateral capacity. In this analysis, the pier is free at the top and the foundation is modeled as discussed earlier. The cross section at the base of the pier is 1.9⫻ 6.1 m. This justifies the higher capacity in the transverse direction 共400%兲 as compared with the longitudinal direction, as shown in Fig. 7. Clearly, the stiffness and capacity of the shorter pier is higher than that of its counterpart. The increase in capacity when the mean material strength is used in analysis is about 9% compared to the case employing characteristic strength values. Analysis of the Bridge in the Transverse Direction Since the bridge is not symmetric, the analysis is carried out in both the positive and negative z-direction. Fig. 8 shows comparison of the sequence of the plastic hinge formation and the capacity envelope of the design and the as-built configurations when applying the incremental load in the z-direction. The idealized capacity envelope of the as-built bridge is also shown. The top displacement is the absolute value at the top of Pier 4, where the maximum drift is observed, while the base shear is the absolute total base shear of all piers. The drift limits 共1, 2, and 3%兲 shown on the graph are based on the height of Piers 2 to 6, which govern the response of the bridge in the transverse direction. The following observations are worth mentioning: • The increase in capacity of the as-built model compared with the design assumption is about 5%. For the design model, which disregards friction, the response is identical when applying the load in the positive and negative z-direction. For the as-built configuration, the capacity in the positive z-direction

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Fig. 7. Capacity envelopes of Piers 1 and 2 in longitudinal and transverse directions using characteristic and mean material strength

is slightly higher 共2%兲 because the sliders contribute frictional resistance of the curved unit. • The first indication of local yielding occurs at the base of Pier 4. This is attained at a total base shear higher than the design level. This indicates that the response will remain in the elastic range under the design earthquake. Global yielding is estimated from the idealized response of the as-built bridge at a drift of 1.3%. This is lower than the design assumption 共1.5%兲, which has lower initial stiffness because it disregards bearing friction. The ultimate capacity is significantly higher than the design shear force. The observed overstrength 共⍀d⫽actual/design strength⫽2.4兲 reflects the reliability of the bridge in this direction 共Elnashai and Mwafy 2002兲. • No degradation in the lateral strength or sway mechanism is observed up to the drift collapse limit state. The pertinent performance indicators in this direction are therefore: 共1兲 the segment collision at abutments; 共2兲 the drift 共1.5% for yielding and 3.0% for collapse兲; and 共3兲 member criteria 共local yielding and ultimate curvature兲. As a result of the curvature of the bridge, the longitudinal collision at abutments is the controlling collapse criterion even when applying the load in the transverse direction.

Analysis of the Bridge in the Longitudinal Direction Longitudinal movement of the superstructure is controlled by the bearing friction. Since the bridge in this direction consists of two units with distinct characteristics, the pushover analysis is separately performed for each unit. For Unit 1, the incremental lateral load is applied in the negative x-direction, it is applied in the positive x-direction for Unit 2 so as to avoid the collision of the two units, as indicated in Fig. 9. Comparisons of the sequence of plastic hinge formation and the capacity envelope of the design and the as-built configurations are shows in Fig. 9 for the two units of the bridge. A comparison between capacity of the design assumption and the as-built bridge—along with the demand, which is discussed in subsequent sections—is depicted in Fig. 10. The following observations are worth noting: • For Unit 1, the capacity envelope of the design configuration is comparable to that obtained from pushover analysis of the individual piers. The ultimate capacity is slightly higher than the design force. The observed overstrength factor 共⍀d兲 is only 1.33. For the as-built configuration, the capacity envelope is higher than that obtained from pushover analysis of the individual piers because of the bearing friction at Abutment A and

Fig. 8. Mapping of sequence of plastic hinge formation, and capacity versus design and demand of design and as-built configurations in transverse direction JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007 / 717

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Fig. 9. Mapping of sequence of plastic hinge formation and capacity envelope of design and as-built configurations in longitudinal direction

at the expansion joint. The ultimate capacity of this unit is therefore higher than that for the design configuration by 15%, while the observed overstrength is 1.5. • For Unit 2, which is 40% longer and has higher gravity loads and masses than Unit 1, the lateral loads of the design configuration are resisted only by Piers 6 and 7. Hence, Unit 2’s lateral force resisting system is significantly less efficient than Unit 1’s. Also, the design base shear of Pier 7 is slightly higher than its ultimate capacity. This is a common problem in bridges crossing steep-sided river valleys. Since the cross sections of the piers are identical, the shorter pier attracts higher forces than the taller one. Unlike those in the design assumption, all piers of the as-built configuration participate in resisting lateral forces. Hence, the ultimate strength of the bridge as built is 50% higher than that of the as-designed structure. The capacity envelopes of Piers 5 and 8, which participate in resisting lateral forces in this modeling, are equivalent to the frictional resistance of the PTFE bearings. Overstrength is 1.4, which represents an observable enhancement over the design configuration. • The overall capacity of the entire bridge increases by 30% as a result of the added resistance from the movable bearings at intermediate piers and at abutments. The lateral capacity in the transverse direction is considerably higher than that in the longitudinal direction 共240% and 180% for the design and the

as-built configuration, respectively兲 because the pier stiffness varies in both directions. No degradation in lateral strength is observed up to a drift of 3%. Global yielding is estimated from the idealized response at a drift of 1.0% and 0.8% for Unit 1 and Unit 2, respectively. The potential sway mechanism of the two units, which involves formation of plastic hinges at the base of Piers 1, 2, 3, 6, and 7, develops at a drift of 1.1%. These limit states are attained at higher deformations than the gap width at the abutments 共101.6 mm⬃ a drift of 0.46%兲. The controlled deformation at the abutments therefore prevents undesirable modes of failure. The collision at abutments is thus the governing performance criterion in the longitudinal direction.

Demand Predictions at the Design Earthquake Before executing the inelastic response history analysis, a number of parameters were investigated to tune the analytical models for this demanding analysis. Significance of the inclination in the coordinate systems 共refer to Fig. 1兲 on the seismic demand predictions is first investigated. Results obtained from applying the seismic loads in both the local and global coordinate systems confirmed the insignificance of this inclination and the applicabil-

Fig. 10. Capacity versus design and demand in longitudinal direction 共design and twice design earthquake兲 718 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007

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Table 2. Inelastic Demands of Design and As-Built Configurations in Transverse Direction from Lowland Records 共kN− mm兲 Demand of the design configuration Design force Location

Lowland1

Lowland2

Demand of the as-built configuration

Lowland3

Lowland1

Lowland2

Lowland3

Displacement Shear Displacement Shear Displacement Shear Displacement Shear Displacement Shear Displacement Shear Displacement Shear

Abutment A

10

Top of Pier 1

17

1721

14

14 3874

7

1820

5

1300

19

4242

6

1729

5

1216

Top of Pier 2

33

2260

54

3898

21

2469

13

1605

53

3939

19

4084

12

2557

Top of Pier 3

42

2811

65

4572

28

2863

17

2145

59

4283

26

3607

17

2215

Top of Pier 4

50

3350

75

5061

48

4275

26

2740

78

5309

48

3702

26

3387

Top of Pier 5

45

3123

59

4168

39

3412

18

2297

54

4134

33

3702

17

2089

Top of Pier 6

35

2415

64

4317

26

3044

15

1701

53

4213

20

2329

12

1470

Top of Pier 7

24

2411

34

3322

10

3893

6

1332

19

3896

10

1989

5

1222

Top of Pier 8

20

2015

41

3794

11

1887

9

1263

24

4062

9

2314

6

1264

Abutment A

11

48

7

8

15

14

Total shear/ 50 20106 75 25556 48 17532 26 Max displacement Note: Demands are at the pier base 共shear兲 and at the pier top 共displacements兲.

ity of using the global coordinate system in analysis. Moreover, since the hysteretic damping caused by inelastic energy absorption is already accounted for in the inelastic analysis, two levels of Rayleigh damping ratios are investigated: 1 and 2%. Comparisons of results of elastic analysis with a 5% damping ratio and ZEUS-NL inelastic response history analyses performed using different Rayleigh damping levels indicated that the 2% Rayleigh damping ratio results improved predictions of deformation and base shear demands, whereas those at the 1% damping are almost twice the elastic analysis results. The Rayleigh damping ratio was maintained at 2% in the inelastic analysis because high inelasticity was expected in the longitudinal direction of the bridge. The integration of the motion equations is also investigated by comparing of results from the Newmark and Hilber-Hughes-Taylor integration algorithms. The results indicate that the latter algorithm is more beneficial under high input ground motions because it can reduce the high short-duration peaks in the solution. Therefore, the Hilber-Hughes-Taylor scheme is employed in response history analyses. Response in Transverse Direction Table 2 shows comparisons between the displacement and base shear demands of the design and the as-built configurations obtained from inelastic response history analysis using the lowland records. Fig. 11 depicts a sample comparison between the maximum displacement time histories at the top of the piers and abutments of the two modeling approaches investigated in this study. The highest demands are generated from Lowland1 at Pier 4. The maximum displacement demands are higher by 50% than those obtained from response spectrum analysis using the design spectrum. This was expected because Lowland1 is higher than the design spectrum in the period range below 1.0 s, as shown in Fig. 5. It is noteworthy that the maximum total base shear demands shown in Table 2 are from the summation of the response time histories and not from summation of the peak values. The predominant cracked periods in the transverse direction are 0.88 and 0.85 s for the design and the as-built models, respectively. These periods are estimated from Fourier analysis of the top acceleration response of Pier 4. The period of the structure in

15

6

29 11294

78

9

13 22140

48

10 16570

26

11239

this direction is marginally influenced by the bearing friction. The maximum demands of the as-built bridge are therefore akin to those observed for the design configuration. Nevertheless, comparison of the displacement time histories shown in Fig. 11 confirms the significant change in response of the curved unit of the bridge 共Piers 5–8 and Abutment B兲. Clearly, the significance of the bearing friction increases with the inclination of the local coordinate system. Hence, it is more pronounced on Abutment B than on Pier 5, and it has no effect on the straight unit 共Abutment A and Piers 1–3兲. Comparisons of the relative displacement in the longitudinal direction at the abutments and the expansion joint for the design and the as-built configurations are depicted in Fig. 12. Although seismic loads are applied in the transverse direction, the response in the longitudinal direction is significantly affected by the bearing modeling approach. For the design configuration, zero deformation demands are observed at Abutment A since Unit 1 of the bridge is straight and the seismic action is applied in the transverse direction. As a result of the curvature of Unit 2, observable displacement demands are generated at the expansion joint and at Abutment B. When the bearing frictional resistance is considered, the two units of the bridge are coupled and the displacement demands are favorably redistributed between the expansion joint and the two abutment gaps. This results in a higher margin of safety, as shown in Fig. 12共b兲. The elongation in period at the design earthquake is about 10% for the two analytical models investigated. No yielding is observed under the effect of the six records employed here, confirming the high margin of safety. A comparison of the capacity and maximum base shear demands of the design assumption and the as-built bridge at the design and twice the design earthquake is shown in Fig. 8. The capacities are estimated from inelastic pushover analysis of the entire bridge and the demands are conservatively estimated from Lowland1, which produced the highest response. It is clear that maximum global demand under the design earthquake is well below the minimum supply and the response of both the design and the as-built configuration is within the elastic range.

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Fig. 11. Comparison of displacement histories in transverse direction at design earthquake 共Lowland1兲

Response in Longitudinal Direction Sample results of the maximum relative displacement and base shear demands from the lowland input ground motions are shown in Table 3. The displacement demands of the as-built configuration generally decrease because of the beneficial effect of bearing friction. In contrast, seismic loads significantly amplify as a result of the considerable reduction in period. Hence, the total base shear demands observed from the earthquake scenarios employed herein are significantly higher 共up to 70%兲 than those from the design assumptions. The maximum shear demand of a number of

piers is higher by up to 110% than the design force. Owing to the high design overstrength, no crushing in concrete is observed. Nevertheless, this significant increase in seismic base shear is a clear indication that neglecting the bearing friction in the design is not conservative. Fig. 13 depicts a comparison of the maximum displacement time histories of the design and the as-built configurations obtained from Lowland1, which produces the maximum demands. The seismic response in this direction is fundamentally different for the two investigated configurations. In the design assumption,

Fig. 12. Comparison of relative displacement response in longitudinal direction at abutments and expansion joint 共load in transverse direction at design earthquake兲 720 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007

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Table 3. Inelastic Demands of Design and As-Built Configurations in Longitudinal Direction from Lowland Records 共kN− mm兲 Demand of the design conf. Design force Location

Abutment A Top of Pier 1 Top of Pier 2 Top of Pier 3 Top of Pier 4 Top of Pier 5 Top of Pier 6 Top of Pier 7 Top of Pier 8 Abutment A

Displacement

61.6 61.5 62.1 62.4 a

80.9 78.2 73.5 68.7 80.6

Lowland1

Lowland2

Demand of the as-built conf.

Lowland3

Lowland1

Lowland2

Lowland3

Shear

Displacement

Shear

Displacement

Shear

Displacement

Shear

Displacement

Shear

Displacement

Shear

Displacement

Shear

3185 1472 1472 1130 1148 1837 4057 1383

108 102 102 103 55 99 90 84 83 80

2389 1291 1249 1330 1345 1537 2257 1501

34 28 35 33 23 49 45 43 44 43

1541 834 872 805 745 928 1593 1304

22 20 25 23 21 26 23 17 21 19

1290 709 658 717 718 648 1087 1301

67 64 63 67 62 58 55 47 44 40

2191 1849 1302 2394 2273 1216 1638 2853

26 26 21 24 31 18 16 16 25 20

1714 1879 1250 1548 1676 773 1549 2837

19 15 21 18 26 19 18 17 16 19

1425 1116 773 2214 2143 779 1256 2292

26

3614

67

10289

31

6261

26

4578

Total shear/ 81.7 15684 108 6197 49 4817 Max displacement Note: Demands are at the pier base 共shear兲 and at the pier top 共displacement兲. a 62.6 for Unit 1 and 81.7 for Unit 2 共at the superstructure level兲.

the two units of the bridge have different dynamic characteristics and seismic response. Hence, the displacement time histories of Unit 1 共Abutment A and Piers 1–3兲 are different from those of Unit 2 共Piers 5–8 and Abutment B兲, as shown in Fig. 13共a兲. Pier 4 at the expansion joint has its own distinct response. In the as-built model, the bearing friction couples the two units of the bridge and stimulates the beneficial contribution of Piers 4, 5, and 8 to the lateral force resisting system. Fig. 13共b兲 clearly shows that the displacement response of Unit 1 is comparable to that of Unit 2, with different amplitudes. Fig. 14 depicts a comparison between Fourier spectra of the two configurations investigated obtained from the top acceleration histories at Pier 1. Clearly, the inelastic periods of the design assumptions are significantly

longer than the as-built bridge. Fig. 14共c兲 pictorially shows the significance of the reduction in period on increasing the seismic loads of the as-built configuration. Comparison of the energy dissipated in different PTFE sliders for the bridge featuring the as-built configuration is shown in Fig. 15. It is clear from the hysteresis loops that the PTFE sliders can dissipate a large amount of energy under the design earthquake. This results in lower deformation demands as compared to those in the design assumption. It is also shown that the high displacement demand at Abutment A controls the response because this abutment has a lower frictional force, which is proportional with the normal stresses. The maximum relative displacement demands at the abutments are shown in Fig. 13. The demand at Abutment

Fig. 13. Comparison of displacement histories in longitudinal direction at design earthquake 共Lowland1兲 JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007 / 721

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Fig. 14. Amplification of seismic forces caused by reduction in period in longitudinal direction: 共a兲 and 共b兲 Fourier spectra of Pier 2 top response for design and as-built bridge; and 共c兲 response spectra of input ground motions

A nearly reaches the supply 共101.6 mm兲 for the design configuration. Also, high demands are observed at Abutment B and at the expansion joint. The PTFE bearing’s frictional resistance significantly reduces these demands in the as-built bridge, particularly at the expansion joint, since the two units of the bridge oscillate simultaneously. Comparisons of the capacity, the design, and the base shear demands of the as-built and the design configurations are shown in Fig. 10共a兲. The total design force estimated from response spectrum analysis is overconservative as it represents the summation of maximum shear forces of various piers, which do not occur simultaneously during the analysis. The realistic summation of base shear time histories obtained from response history analysis produces significantly lower demands. For the two analytical idealizations investigated, the minimum supply is higher than the maximum demand, which is, in turn, lower than the design. It is clear from the comprehensive capacity-demand comparisons presented above that the performance of the design assumption in the longitudinal direction is satisfied at the design earthquake for all limit states except for the collision at abutments. This confirms the conclusions from pushover analysis discussed earlier. Including the frictional resistance of PTFE bearings redistributes the displacement demands to tolerable limits. The performance of the as-built bridge in this direction is therefore satisfactory for all limit states. Nevertheless, it is important to stress that the investigated bridge exhibits high overstrength because the design was governed by flow and ice loads. Therefore, the amplification of

seismic forces and base shear demands of the as-built bridge confirms the high uncertainties arising from the assumptions typically used in the design of complex bridges.

Demand Predictions at Twice the Design Earthquake The results shown at the design earthquake indicated that the maximum demands are generated under the effect of the Blytheville earthquake scenario. The accelerograms generated for this scenario were therefore scaled up to twice the design ground motion and employed to investigate the vulnerability of the bridge under the most credible earthquake. Comparisons between the maximum inelastic demands of the design and the as-built configurations at twice the design earthquake are shown in Table 4. Response in Transverse Direction The effect of bearing friction is more pronounced at this high level of ground motion. The base shear demands of the as-built behavior are higher than those in the as-designed structure. Maximum displacement demands are 100% higher than those observed at the design earthquake and 300% higher than those used in the design. The maximum observed drift is 0.80%, which is quite satisfactory. The maximum relative displacement demands in the longitudinal direction at the expansion joint and abutments 共gap

Fig. 15. Energy dissipated in PTFE sliders for bridge featuring as-built configuration at design earthquake 722 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007

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Table 4. Comparison between Maximum Inelastic Displacement and Base Shear Demands of Design and As-Built Configuration at Twice Design Ground Motion 共kN− mm兲 Transverse direction

Longitudinal direction As-built

Design assumptions Location Abutment A Top of Pier 1 Top of Pier 2 Top of Pier 3 Top of Pier 4 Top of Pier 5 Top of Pier 6 Top of Pier 7 Top of Pier 8 Abutment B

Design assumptions

As-built

a

Without gaps

0.0

76

5007 8555 5875 7564 5926 5795 5506 5831

53

29 32 89 105 158 97 92 44 48 42

a

With gaps

Displacement Gaps Shear Displacement Gaps Shear Displacement Gaps Shear Displacement Gaps Shear Displacement Gapsa Shear 24 27 79 92 130 96 90 49 52 58

a

17

17

5797 9182 6198 8750 5947 6463 6159 6982

26

234 226 226 228 127 147 140 125 126 116

191 3895 3173 2024 187 1754 1538 1631 1876 1945 116

a

183 176 189 188 133 139 130 121 115 108

Total shear/ 130 33960 158 35539 234 11835 189 Max displacement a Maximum relative displacement 共gap close兲 in the longitudinal direction at the expansion joint and abutments.

close兲 are given in Table 4. It is clear that no collision is detected and the margin of safety is satisfactory at this high level of ground motion. A comparison of the global capacity to the maximum base shear demand at this level of ground motion is shown in Fig. 8. It is clear that limited inelasticity is generated and the demands are well below the minimum supply. The results confirm the satisfactory performance and the adequate margin of safety in the transverse direction at twice the design ground motion. Response in Longitudinal Direction Comparison of the inelastic demands of the design to those of the as-built bridge is shown in Table 4. The analysis was conducted for the as-built bridge with and without modeling the structural gaps at the abutments and expansion joint to highlight the gaps’ significance in the bridge’s seismic response. The inelastic demands in this direction significantly increase at twice the design earthquake. Although base shear demands exceed the design forces for the majority of substructural elements, no crushing in the concrete core is observed because of the as-built bridge’s high overstrength. The response of the design assumption is unacceptable because of its high displacement demands, almost twice the gap width capacity at Abutment A. The as-built structure performed adequately at the design earthquake. Bearing friction effectively coupled the two units of the bridge and significantly improved the seismic capacity in this direction. The response time histories in Fig. 16 clearly show that the two units of the as-built bridge simultaneously vibrate at twice the design PGA. This confirms the significant effect of bearing friction. Modeling the segment collision at the structural gaps reduces the displacement demands and slightly increases the base shear of few piers, as shown from Table 4. Fig. 17 compares the relative displacement response with and without modeling the gaps at the abutments and expansion joint. The results confirm the significance of seismic gaps modeling on the response of multispan curved bridges. Comparisons of the capacity to demand ratios of the pinned and PTFE bearings indicate the acceptable response up to twice

183

74

3536 3272 2007 1653 1497 1516 2925 1963

181 172 171 176 140 127 122 115 115 105

15683

181

108

102

57

3525 3397 2172 1652 1529 1713 2906 1924

88 16016

the design earthquake. For pinned bearings, the superstructure inertia forces are transferred to the substructure through anchor bolts at the top of the piers. The tension and shear capacities of these bolts are 387 and 238 kN, respectively. The maximum tension and shear demands on the bolt are 303 and 176.7 kN, respectively, which are lower than the capacities. For the PTFE bearings, the maximum seismic demand is 4139 kN, which produces a shear strain of 1.8. This results in a total shear strain lower than the allowable limit recommended by seismic codes and guidelines 共NCHRP 12-49 2001兲. Also, the maximum displacement demand at the PTFE bearings in the longitudinal direction is lower than the sole plate limits. Hence, no damage is anticipated to the PTFE surface. Comparison of the overall capacity and maximum base shear demand for the design and the as-built bridge is shown in Fig. 10共b兲. For both configurations, the global demand at twice the design ground motion is slightly below the ultimate capacity. Although controlling the deformation by seismic gaps prevents formation of undesirable modes of failure, it also increases base shear demands. The performance of the as-built bridge in the longitudinal direction is therefore satisfactory for all limit states with the exception of the collision at abutments. The results support the conclusions drawn earlier and emphasize the significance of pushover analysis procedure in identifying potential structural deficiencies and prioritizing limit state criteria for complex bridges.

Conclusions A multispan curved bridge was selected to investigate the influence of frequently used design assumptions on the seismic integrity of complex highway bridges. As opposed to undertaking extensive parametric studies in which the parameters are varied not necessarily according to design specifications, this work focused in intricate detailing of one case of a realistic, office-designed and checked complex bridge. Refined 3D fiber modeling approaches

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Fig. 16. Response in longitudinal direction at twice design earthquake: 共a兲 mapping of plastic hinges for the bridge featuring as-built configuration; 共b兲 and 共c兲 displacement histories for design and as-built bridge

were verified and employed to compare the elastic and inelastic behavior of the as-designed and as-built configurations. The adopted methodology and results of the comprehensive analysis performed to estimate the capacities and predict of demands were presented. The following conclusions were drawn based on the work reported above: • In the transverse direction, the response was less affected by friction because the PTFE sliders were restrained by girder stops. This was more pronounced on the curved unit, particularly when increasing the inclination of substructural members. Overstrength was 2.4 and first yielding was attained at a base shear higher than the design value, reflecting a high safety margin and the anticipated elastic response under the design earthquake. Pushover analysis indicated that the collision at abutments was the controlling performance criterion even when seismic loads were applied in the transverse direc-

tion. Inelastic response history analyses confirmed that the displacement demands at abutments and the expansion joint were significantly affected by friction because of the coupling between the two units of the bridge in the longitudinal direction. • In the longitudinal direction, the modes of vibration were significantly and fundamentally different and the predominant period decreased by 50% when the bearing frictional resistance was accounted for. To avoid collision between the two units of the bridge, pushover analysis was performed independently for Unit 1 and 2 by applying the incremental load in two opposite directions. For the design configuration, Unit 2 was less efficient than Unit 1 because its PTFE piers were excluded from resisting lateral loads. Overstrength of Unit 2 was almost unity, confirming its high vulnerability. For the as-built configuration, all piers participated in resisting lateral forces and the capacity was significantly higher than the as-designed

Fig. 17. Comparison of relative displacement response in longitudinal direction at expansion joint and abutments with and without modeling of gaps 共load in longitudinal direction at twice design earthquake兲 724 / JOURNAL OF BRIDGE ENGINEERING © ASCE / NOVEMBER/DECEMBER 2007

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structure. Pushover analysis confirmed that the superstructureabutment zones controlled the response. The displacement demands of the design assumption were alarming at the abutments, but they were satisfactory for the as-built bridge. Seismic loads increased because the period was considerably reduced, which, in turn, magnified the base shear demands of the as-built configuration 共up to 110%兲. The results clearly highlighted the uncertainties arising from the assumptions typically used in the design of complex bridges. • At twice the design earthquake, the safety margins of the asbuilt bridge in the transverse direction were satisfactory, and the structure was just at the onset of the post-elastic range. In the longitudinal direction, the displacement demands increased significantly, while base shear demands exceeded the design forces for the majority of substructural members. It was confirmed that bearing friction maintained the coupling between the two units of the bridge. Modeling of structural gaps was significant in controlling the displacement demands, which resulted in limited inelasticity at the high level of ground motion. The performance of the as-built bridge in the longitudinal direction was therefore satisfactory for all limit states, except for the collision at abutments. The study emphasized the importance of pushover analysis procedures in verification of analytical models, identifying potential structural deficiencies, estimating capacity, and providing insight into the limit states of multispan complex bridges. The significance of simplifying design assumptions on the inelastic seismic response of the multispan complex bridge was confirmed. The results adequately represent other complex bridges. The dynamic characteristics fundamentally changed and the capacity increased when friction was included. The latter change impacted the seismic demand adversely by significantly reducing the periods of vibration. Assumptions aimed at simplifying the design process may therefore compromise the safety of bridges similar to that studied here. Structural gaps at the abutments controlled the response of the curved bridge in the two principle directions. Attention should be focused, therefore, on characterizing the behavior of joints and their effect on design actions and deformations.

Acknowledgments This study was funded by the U.S. Federal Highway Administration 共FHWA兲 through the Mid-America Earthquake 共MAE兲 Center, University of Illinois at Urbana-Champaign. The MAE Center is an engineering research center funded by the National Science Foundation under cooperative agreement reference EEC 97-01785. The writers are grateful to Professor Glenn Rix, Georgia Institute of Technology, for his advice on earthquake input motions.

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